Shading - How do I remove overlapping seams in X- and Y-direction while using vectors?
As I wrote in the comments, some seams are inevitable. Even on a cube, if a side is 1.1 m long, and a tile is 0.2 m long, then 5 tiles will fit completely and half of a tile will be cut off in middle (giving an impression of a seam). What if you don't have a cube? Any non-vertical face will be visible from side as a diagonal line, which will not be compatible with squares that you look at en face.
Another way to look at it is this: if you were able to procedurally draw squares on an arbitrary figure without seams, it would mean you tessellate that figure with squares, which I think is fair to say is impossible - spheres in Blender are made either from triangles or from heavily stretched quads (and some triangles on top and bottom if you don't cast a subdivided cube into a sphere).
So while some seams (on the left) seem to be unavoidable, the seams inside NSWE, top, bottom sides can be avoided (on the right):

I came up with a basic setup like this:

By increasing Epsilons, you remove the inner seams by removing the normal-aligned axis information from the vector:

You will still get seams on areas between those &quot;flattened&quot; areas.
You could improve that node setup, or maybe instead create a black texture and use R/G/B information, and draw on that texture:

Best to draw in Add mode to add flattening (use the color of an axis visible on the gizmo: X:red, Y:green, Z:blue), and Subtract mode to remove flattening. I'd recommend falloff set to constant.
Perhaps a better idea is to make that information only offset the coordinates, so you're not forced to draw on an entire side of the object, and can just draw on a part of a square on one side of a seam.